Informationally complete sets of Gaussian measurements

TL;DR

This paper establishes the necessary and sufficient conditions for the informational completeness of Gaussian measurements in continuous variable quantum systems, detailing the structure and minimal requirements for such measurement sets.

Contribution

It provides a comprehensive characterization of informationally complete Gaussian measurement sets, including conditions for single and multiple observables, and extends to general linear bosonic channels.

Findings

Single informationally complete observable has phase space as outcome space

Infinite observables are required for projection valued Gaussian measurements

Characterization of informational completeness for general linear bosonic channels

Abstract

We prove necessary and sufficient conditions for the informational completeness of an arbitrary set of Gaussian observables on continuous variable systems with finite number of degrees of freedom. In particular, we show that an informationally complete set either contains a single informationally complete observable, or includes infinitely many observables. We show that for a single informationally complete observable, the minimal outcome space is the phase space, and the observable can always be obtained from the quantum optical $Q$-function by linear postprocessing and Gaussian convolution, in a suitable symplectic coordinatization of the phase space. In the case of projection valued Gaussian observables, e.g., generalized field quadratures, we show that an informationally complete set of observables is necessarily infinite. Finally, we generalize the treatment to the case where the measurement coupling is given by a general linear bosonic channel, and characterize informational completeness for an arbitrary set of the associated observables.